2 These authors contributed equally to this work.While neural networks have been remarkably successful for a variety of practical problems, they are often applied as a black box, which limits their utility for scientific discoveries. Here, we present a neural network architecture that can be used to discover physical concepts from experimental data without being provided with additional prior knowledge. For a variety of simple systems in classical and quantum mechanics, our network learns to compress experimental data to a simple representation and uses the representation to answer questions about the physical system. Physical concepts can be extracted from the learned representation, namely: (1) The representation stores the physically relevant parameters, like the frequency of a pendulum.(2) The network finds and exploits conservation laws: it stores the total angular momentum to predict the motion of two colliding particles. (3) Given measurement data of a simple quantum mechanical system, the network correctly recognizes the number of degrees of freedom describing the underlying quantum state. (4) Given a time series of the positions of the Sun and Mars as observed from Earth, the network discovers the heliocentric model of the solar systemthat is, it encodes the data into the angles of the two planets as seen from the Sun. Our work provides a first step towards answering the question whether the traditional ways by which physicists model nature naturally arise from the experimental data without any mathematical and physical pre-knowledge, or if there are alternative elegant formalisms, which may solve some of the fundamental conceptual problems in modern physics, such as the measurement problem in quantum mechanics.Problem: Predict the position of a one-dimensional damped pendulum at different times. Physical model: Equation of motionSolution:Observation: Time series of positions: o = x(t i ) i∈{1,...,50} ∈ R 50 , with equally spaced t i . Mass m = 1kg, amplitude A 0 = 1m and phase δ 0 = 0 are fixed; spring constant κ ∈ [5, 10] kg/s 2 and damping factor b ∈ [0.5, 1] kg/s are varied between training samples. Question: Prediction times: q = t pred ∈ R.Correct answer: Position at time t pred : a cor = x(t pred ) ∈ R .Implementation: Network depicted in Figure 1b with 3 latent neurons. Key findings:• SciNet predicts the positions x(t pred ) with a root mean square error below 2% (with respect to the amplitude A 0 = 1m) (Figure 2a).• SciNet stores κ and b in two of the latent neurons, and does not store any information in the third latent neuron (Figure 2b).

We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.

We consider the decomposition of arbitrary isometries into a sequence of single-qubit and Controlled-not (C-not) gates. In many experimental architectures, the C-not gate is relatively 'expensive' and hence we aim to keep the number of these as low as possible. We derive a theoretical lower bound on the number of C-not gates required to decompose an arbitrary isometry from m to n qubits, and give three explicit gate decompositions that achieve this bound up to a factor of about two in the leading order. We also perform some bespoke optimizations for certain cases where m and n are small. In addition, we show how to apply our result for isometries to give a decomposition scheme for an arbitrary quantum operation via Stinespring's theorem, and derive a lower bound on the number of C-nots in this case too. These results will have an impact on experimental efforts to build a quantum computer, enabling them to go further with the same resources.